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This paper presents a unified framework for smooth convex regularization ofdiscrete optimal transport problems. In this context, the regularized optimaltransport turns out to be equivalent to a matrix nearness problem with respectto Bregman divergences. Our framework thus naturally generalizes a previouslyproposed regularization based on the Boltzmann-Shannon entropy related to theKullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. Wecall the regularized optimal transport distance the rot mover's distance inreference to the classical earth mover's distance. We develop two genericschemes that we respectively call the alternate scaling algorithm and thenon-negative alternate scaling algorithm, to compute efficiently theregularized optimal plans depending on whether the domain of the regularizerlies within the non-negative orthant or not. These schemes are based onDykstra's algorithm with alternate Bregman projections, and further exploit theNewton-Raphson method when applied to separable divergences. We enhance theseparable case with a sparse extension to deal with high data dimensions. Wealso instantiate our proposed framework and discuss the inherent specificitiesfor well-known regularizers and statistical divergences in the machine learningand information geometry communities. Finally, we demonstrate the merits of ourmethods with experiments using synthetic data to illustrate the effect ofdifferent regularizers and penalties on the solutions, as well as real-worlddata for a pattern recognition application to audio scene classification.

Regularized Optimal Transport and the Rot Mover's Distance